It is written in the book that if $h(x)$ is convex then $f(x)=h(Ax+b)$ is also convex. Now according to the composition rules if we write $f(x)=h(g(x))$ where $g(x)=Ax+b$ then if
$h(x)$ is convex and nondecreasing and $g(x)$ is also convex then $f(x)$ will be convex. Although $g(x)$ is convex here, due to being affine, and $h(x)$ is convex by assumption but how can we be sure that $h(x)$ is nondecreasing?
$h(x)$ is convex and non-increasing and $g(x)$ is concave then $f(x)$ will be convex. Although $g(x)$ is concave here, due to being affine, and $h(x)$ is convex by assumption but how can we be sure that $h(x)$ is nonincreasing?
Any help in this regard will be much appreciated. Thanks in advance.